Stat 694 Lecture Notes -- Thurs., 3/2/06
Lecture – Thurs., 3/2/06
Today • Finish seeds example – Bayesian logistic regression • Overdispersed generalized linear models • PHDCN example – Bayesian hierarchical logistic regression model • Bayesian generalized linear model for count data – Poisson (log- linear) regression model • Salmonella example
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Bayesian Generalized Linear Model
GENERALIZED LINEAR MODEL
Idea: extend the idea of linear modeling to cases where either 1. the relationship between X and E(y|X ) is not linear 2. assumption of a normal sampling distribution is inappropriate
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1 Stat 694 Lecture Notes -- Thurs., 3/2/06
Bayesian Generalized Linear Model SEEDS EXAMPLE • We want to look at the relationship between germination and both seed type (73 or 75) and root extract (bean or cucumber).
LOGISTIC REGRESSION MODEL
p(r i|p i,n i) = Bin(p i,n i), where
logit(p i) = α0+α1 xi1 + α2xi2 +α12 xi1 xi2
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Bayesian Generalized Linear Model
BAYESIAN LOGISTIC REGRESSION MODEL
• Likelihood
• Prior on the αs
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2 Stat 694 Lecture Notes -- Thurs., 3/2/06
Bayesian Generalized Linear Model Definition We say that a generalized linear model is overdispersed if the model allows the possibility of variation beyond that of the assumed sampling distribution.
Bayesian Overdispersed (Random Effects) Logistic Regression Model
• p(r i|p i,n i) = Bin(p i,n i), where
logit(p i) = α0+α1 x1 + α2x2 +α12 x1x2+b i
2 2 • p(b i|σ ) = N(0, σ )
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Bayesian Generalized Linear Model
PROJECT ON HUMAN DEVELOPMENT (PHDCN) EXAMPLE
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3 Stat 694 Lecture Notes -- Thurs., 3/2/06
Bayesian Generalized Linear Model
RESPONSE VARIABLE • BMI
INDIVIDAL COVARIATES • age1 • famsize • sescomp • gender • fams1 • latino • black
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Bayesian Generalized Linear Model
NC COVARIATES • pclnpov
• pcstab3
• comstrt8
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4 Stat 694 Lecture Notes -- Thurs., 3/2/06
Bayesian Generalized Linear Model
HIERARCHICAL LOGISTIC REGRESSION MODEL
For the ith student in the jth neighborhood, (I) • p(Y i,j |p i,j ) = Bern(p i,j ), where logit(p i,j ) = Xi,j β + µj
(N) • µj = αj + Zjβ
• p( αj) = • p( β(I))= • p( β(N))=
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Bayesian Generalized Linear Model
POSTERIOR SUMMARIES
Variable Post. Mean (95% Credible Interval) intrcpt -4.106 (-5.112,-2.993) age1 0.149 (0.077,0.21) famsize -0.008 (-0.091,0.069) sescomp -0.128 (-0.257,-0.012) gender 0.123 (-0.209,0.445) fams1 -0.004 (-0.356,0.375) latino 0.418 (-0.052,0.931) black 0.792 (0.294,1.374)
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5 Stat 694 Lecture Notes -- Thurs., 3/2/06
Bayesian Generalized Linear Model
POSTERIOR SUMMARIES
Variable Post. Mean (95% Credible Interval) pclnpov 0.017 (-0.171,0.212) pcstab3 0.193 (-0.035,0.415) cmstrt8 0.099 (-0.124,0.325)
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Bayesian Generalized Linear Model
POSTERIOR SUMMARIES
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6 Stat 694 Lecture Notes -- Thurs., 3/2/06
Bayesian Generalized Linear Model
GENERALIZED LINEAR MODEL – COUNT DATA Assumption:
SALMONELLA EXAMPLE (Breslow, 1984)
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Bayesian Generalized Linear Model
OVERDISPERSED POISSON REGRESSION MODEL
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